[R-SIG-Finance] Generating Distributions with set skewness and kurtosis

[Matthew Clegg]

On Tue, Aug 26, 2008 at 11:53 AM,  <Matthias.Koberstein at hsbctrinkaus.de> wrote:
>
> Hello,
>
> I am reaching out to you for help since I am struggeling to find a function
> to generate distributions with a set statistical properties as kurtosis and
> skewdness.
> Lets say I want to generate random variables following a "normal"
> distribution, but with skewness 2 and kurtosis 5.
> How would I do that, the most efficient way? Are there any packages for
> that? I had a quick look but were only able to find packages which
> calculate statistical
> distribution properties after having the data.
>
> Thank you very much
>
> Matthias
>

The skewness and kurtosis of the normal distribution are fixed,
but there are many continuous univariate distributions defined
on the entire real line for which the skewness and kurtosis can be
varied.

One possible choice is the Pearson Type IV distribution.
This distribution has the nice feature that the skewness and
kurtosis can be easily formulated in terms of the distributional
parameters (and vice versa).  The Wikipedia entry of the Pearson
distributions is fairly informative:

http://en.wikipedia.org/wiki/Pearson_distribution

Joel Heinrich has written up a nice implementation guide:

http://www-cdf.fnal.gov/publications/cdf6820_pearson4.pdf

The translation into R is fairly straightforward.

There are many other options for distributions that allow for
arbitrary skewness and kurtosis, but relating the parameters of the
distribution to the skewness and kurtosis can be a challenge.
If you are willing to resort to numerical methods to determine
the skewness and kurtosis from the distributional parameters,
here are a few choices.

One easy option is the skewed-t distribution of Férnandez and Steel.
See the "skewt" package by Robert King and Emily Anderson.  The
Férnandez and Steel approach is elegant in that it provides a way to
transform any symmetric continuous distribution into a skewed distribution.
However, working out the exact skewness and kurtosis from the
parameter values can be a challenge.

As mentioned by John Frain, the stable distribution is a good
choice when the tails are especially heavy.  See John Nolan's
web site for a wealth of information:

http://academic2.american.edu/~jpnolan/stable/stable.html

For an R implementation, see Jim Lindsey's web page:

http://popgen.unimaas.nl/~jlindsey/rcode.html

(Also, although the stable distributions are skewed and
heavy-tailed, the traditional definitions of skewness and kurtosis
can't be applied to them, because the 2nd and higher moments
are not defined.)

On the other hand, if you are interested in a distribution that
has thinner tails than the normal, you might want to consider
the skew GED distribution.  See Diethelm Wuertz's fGarch
package:

http://www.rmetrics.org

This package also contains implementations of skew normal
and skew student-t distributions, again using Férnandez and
Steel's approach.

The normal inverse Gaussian, and its cousin the generalized
hyperbolic distribution, has received a fair amount of recent
attention.  I believe an implementation can be found in the
"ghyp" package of Wolfgang Breymann and David Luethi.

This does not by any means exhaust the space of possibilities,
but it should at least give you a start.

BTW, here are a few R functions that will help you to explore
the skewness and kurtosis of arbitrary distributions:

# Calculate mean of an arbitrary density
Mean <- function(f, ...) { integrate(function (x) { f(x, ...) * x },
-Inf, Inf)$value }
# Calculate k-th central moment of an arbitrary density
M <- function (f, ..., k=1, xm = Mean(f, ...)) { integrate(function(x)
{(x - xm)^k*f(x,...)}, -Inf, Inf)$value }
# Calculate skewness of an arbitrary density
SK <- function(f, ...) { M(f, ..., k=3) / (M(f, ..., k=2)^1.5) }
# Calculate excess kurtosis of an arbitrary density
KU <- function(f, ...) { M(f, ..., k=4) / (M(f, ..., k=2)^2) - 3}

> SK(dnorm)
[1] 0   # normal distribution has skewness of 0
> KU(dnorm)
[1] 1.625367e-13  # good enough for government work

# Test gamma distribution with shape=1
> Mean(dgamma, 1)
[1] 1  # good
> SK(dgamma, 1)
[1] 2  # good
> KU(dgamma, 1)
[1] 6  # good

> library(skewt)
> KU(dskt, 5, 1)
[1] 6   # Agrees with theory ... skewness of t with 5 d.f. should be 6
> SK(dskt, 5, 1.5)
[1] 1.516366
> SK(dskt, 5, 1/1.5)
[1] -1.516366


-- 
Matt Clegg
matthewcleggphd at gmail.com